J. Phys. Chem. B 2008, 112, 703-709
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Dynamics of Water in NaxCoO2‚yH2O Niina Jalarvo,† Heloisa N. Bordallo,*,†,‡ Nadir Aliouane,† Mark A. Adams,§ Jo1 rg Pieper,†,| and Dimitri N. Argyriou† Hahn-Meitner Institut, Glienicker Strasse 100, 14109 Berlin, Germany, Institut Laue LangeVin, B.P. 156, 38042 Grenoble, Cedex 9, France, ISIS Pulsed Neutron Scattering Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, U.K., and Technische UniVersita¨t Berlin, Institut fu¨r Chemie, Max-Volmer-Laboratorium fu¨r Biophysikalische Chemie, Strasse des 17. Juni 135 10623 Berlin, Germany ReceiVed: June 7, 2007; In Final Form: October 17, 2007
Incoherent inelastic neutron scattering experiments were performed on Na0.7CoO2 and Na0.28CoO2‚1.3H2O in order to understand how the dynamics of the hydrogen-bond network of water is modified in the triangular crystalline lattice NaxCoO2‚yH2O. Using quasi-elastic neutron scattering (QENS), we were able to differentiate between two types of proton dynamics: a fast process (due to water strongly bound into the sodium cobalt oxyhydrate structure during the hydration process) and a slow process (likely attributable to a collective motion). High-resolution QENS experiments, carried out on Na0.28CoO2‚1.3H2O, show that, at temperatures above 310 K, the water dynamics can be well-described by a random jump diffusion model characterized by a diffusion constant equal to 0.9 × 10-9m2/ s, which is significantly lower than the rate of diffusion for bulk water. Furthermore, our results indicate that, at room temperature, the sodium ions have no influence on the rotational dynamics of the “fast” water molecules.
I. Introduction The layered NaxCoO2 system has attracted significant interest in materials science, solid-state physics, and chemistry over the past 20 years. Initially, these compounds were examined because of their technologically important properties as battery materials due to their fast-ionic conductivity1 and, more recently, because of their thermoelectric properties.2 The discovery of novel 5 K superconductivity3 in NaxCoO2‚yH2O, with x ) 0.35 and y ) 1.4, has provided an additional focus for their investigation. In terms of physics, these materials are interesting, as the CoO2 layer forms a quasi-two-dimensional triangular lattice network in which electronic and magnetic interactions are inherently frustrated. Although much discussion has focused on the physical mechanism of superconductivity, the crystalchemical aspects of this superconductor are complex and have not been completely understood.4 Anhydrous NaxCoO2 crystallizes in a hexagonal crystal structure (P63/mmc) in which the Na+ ions order between the CoO2 layers to form long-periodicity superstructures.5 Solid-state methods can be used to produce single-phase samples, as assessed by X-ray diffraction, for x > 0.7, while lower x compositions can be synthesized by deintercalation of Na from a high x composition either via wet chemical or electrochemical means.5,6 This leads typically to a reordering of the Na-ion sublattice to form new crystallographic phases.5,6 The hydration of NaxCoO2 is possible once sufficient Na ions have been removed, typically over the range 0.25 < x < 0.45. Although there are a number of contradictory neutron diffraction investigationsofthecrystalstructureofthehydratesuperconductor,7-9 there is general agreement that the water is inserted to form * Corresponding author. E-mail:
[email protected]. † Hahn-Meitner Institut. ‡ Institut Laue Langevin. § Rutherford Appleton Laboratory. | Technische Universita ¨ t Berlin.
layers between the Na and CoO2 sheets, thus expanding the c axis from ∼11 Å to ∼19.4 Å. The actual structural coordination of Na and H2O has been initially difficult to determine because of the apparent disorder of H2O molecules within the host lattice. However, the combination of neutron and electron diffraction has revealed that the superconducting (SC) phase forms a superstructure with dimensions of 2ahex × 2ahex × chex with respect to the parent P63/mmc hexagonal phase.9 Analysis of the neutron diffraction data based on this supercell has shown that the coordination of Na to H2O is triangular prismatic in the ideal case, as shown in Figure 1, while the ratio of Na to H2O is between 4 and 5, on the basis of the chemical composition of this phase.10 Indeed, these results were later confirmed by single-crystal neutron diffraction work by Moyoshi et al.11 These diffraction measurements show, in essence, that the behavior of H2O in the host lattice of this SC hydrate mimics the behavior of water in alkali-water solutions. It is well-known that, on average, six H2O molecules are bound to a Na+ center to form a hydration shell. In such arrangements, the water molecule tends to orient itself so that its polarized charge concentration faces the opposite charge of the ion. The orientation of the molecules in the hydration shell will in turn result in a net charge, with the sign of the ion in the center being on the outside of this shell as well. Normally, this charge located outside the hydration shell tends to orient water molecules in their vicinity, leading to a second hydration shell. The consequence of forming hydration shells is to weaken the structure of the water hydrogen bond network.12 In our particular case, the structure naturally constrains the formation of hydration shells, and only one hydration shell can comfortably be formed around a single Na+ ion. The water protons that describe this state can be regarded as (1) protons that are chemically bound into the sodium cobalt oxyhydrate structure during the hydration
10.1021/jp074398y CCC: $40.75 © 2008 American Chemical Society Published on Web 12/20/2007
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Figure 1. Three-dimensional representation of the structure of NaxCoO2‚yH2O obtained from our Rietveld analysis. For simplicity, only the Na(1) is shown. The NaO3 and HO(Co) bond lengths are also given.
(the so-called bound-water) and (2) protons that can hydrogenbond less strongly in the hydration shell around the Na ions. In this paper, we present incoherent inelastic neutron scattering (IINS) measurements that directly probe the dynamics of the intercalated water in the hydrated SC NaxCoO2‚yH2O. We find that, for T e 310 K, the proton dynamics is characterized only by rotational motion. Translational motion is observed above 310 K, when the thermal energy is sufficient to allow hydrogen bonds to break. Fixed window energy scans indicate a freezing of proton dynamics below 150 K on a tenths of a picosecond time scale. II. Experimental For the two IINS experiments described below, two separate SC polycrystalline samples of NaxCoO2‚yH2O, plus a polycrystalline sample of Na0.7CoO2, were prepared using solid-state chemical synthesis as described in references 4 and 13. Further characterization of the freeze-dried SC samples was done by neutron activation analysis (NAA) and X-ray powder diffraction. NAA was performed to obtain the Na/Co ratio for both SC samples, and was found to be 0.3(2) (NEAT sample) and 0.29(2) (IRIS sample). The water content was estimated to be y ∼ 1.3. X-ray diffraction showed that both samples were single phase with lattice constants ahex ∼ 2.84 Å and chex ∼ 19.4 Å. Neutron Scattering Measurements Using NEAT. Timeof-flight (ToF) neutron scattering spectra from both hydrous and anhydrous samples were measured at 295 K using the multichopper spectrometer NEAT,14 located at the Hahn Meitner Institut (HMI). Here we used incident neutron energies of 3.15 and 1.25 meV (λ ) 5.1 Å and 8.1 Å, giving resolutions of 98 and 30 µeV (full width at half-maximum (fwhm)), respectively, at the elastic peak position within an angular range of 13.3° e φ e 136.7°. Typically, as-prepared samples can contain a certain amount of surface water. Thus, before carrying out the neutron scattering measurements reported here, the SC samples were freeze-dried. The NEAT sample was cooled ex-situ by immersion in a cold bath to a temperature of 270 K in a glass bulb, while the bulb was evacuated by a roughing pump. After 6 h, the sample was placed in a sample can and cooled on the E9 diffractometer (also located at the HMI); a diffraction scan revealed the absence of free water and Bragg peaks from the SC phase only. At the same time, a 200 mg portion of the sample was checked using a Quantum Design MMPS SQUID magnetometer. As demonstrated by the temperature dependence of the magnetic susceptibility χ(T), the sample showed a diamagnetic transition indicative of superconductivity at Tc ∼ 4.8 K (Figure 2).
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Figure 2. Plot of χ(T) indicating that NaxCoO2‚yH2O (x ∼ 0.29 or 0.3, y ∼ 1.3) becomes a superconductor below 4.8 K.
The measured NEAT spectra were corrected for detector efficiency and sample-geometry-dependent attenuation using the program FITMO.15 Additionally, the background was subtracted using a measurement of the empty sample container, and the sample Bragg peaks were removed. Following a classical treatment, the generalized density of states (GDOS)16 was obtained. For the quasi-elastic neutron scattering (QENS) data analysis, the data were grouped to obtain constant-angle spectra, resulting in a wave-vector transfer range of 0.5 e Q e 2.07 Å-1. During the NEAT measurements, the sample holders were placed at 135° in relation to the direction of the incident neutron beam, so the last 10 detectors were not used in the data analysis. Neutron Scattering Measurements Using IRIS. Additional ToF high-resolution experiments were carried out on an SC sample, using the high-resolution neutron backscattering spectrometer IRIS at the ISIS facility, Rutherford Laboratory.17 Our measurements were performed using the 002 reflection of the pyrolytic graphite analyzer bank, with an elastic energy resolution of 17.5 µeV (fwhm). The instrument was run in an offset configuration to give an energy transfer window of -0.2 e ∆E e 1.2 meV and a momentum transfer range of 0.4 e Q e 1.8 Å-1. For this IINS experiment, the sample was freeze-dried insitu at 265 K. Given that IRIS has a diffraction detector bank at 2θ ∼ 170°, diffraction patterns were measured during this process to monitor the disappearance of Bragg reflections due to ice. The freeze-drying process was terminated when ice peaks could no longer be detected. The temperature dependence of χ(T), measured after the experiment, also showed an SC behavior with Tc ∼ 4.8 K. At the same time, monitoring of the (002) reflection at 19.6(2) Å of the SC phase ensured that the sample did not dehydrate even at 320 K. Measured IRIS spectra were corrected in a similar manner to that described above, using the program MODES.18 For the QENS data analysis, after removal of the Bragg peaks, the data were grouped covering a wave-vector transfer range of 0.46 e Q e 1.8 Å-1. III. Results Our neutron scattering experiments probed the proton dynamics of the SC cobaltates in the 0.1-100 ps range. The use of H, which possesses a larger incoherent scattering cross section (binc(H) ∼ 80 barns), ensured that the Sinc (Q,ω) measured here was dominated by the movement of protons.19 Furthermore, in the evaluation of the vibrational spectra for Na0.3CoO2‚1.3H2O, the water-cation stretch modes are expected to be observed, as they contain contributions from both water-cation and H-bond stretches (defined as the stretching of the hydrogen-
Dynamics of H2O in NaxCoO2‚yH2O
J. Phys. Chem. B, Vol. 112, No. 3, 2008 705 TABLE 1: EISF for Rotational Motions in NaxCoO2‚yH2O model
EISF
isotropic two-site jump model
(sin(Qr)Qr)2 12 (1 + sin(Qd)Qd)
jump vector (Å) r ) 0.98 d ) 2.1
TABLE 2: Parameters Characterizing the Rotational and Translational Motion of Water in Na0.28CoO2‚1.3H2Oa T (K)
Dt (10-9 m2/s)
τ0 (ps)
295 310 320 bulk water at 295 K
0.9 ( 0.3 2.49 ( 0.07
21 ( 2 1.57 ( 0.12
τr (ps) 1.85 (fast) 20 (slow) 22 (slow) 1.05
a
Values were obtained at 295 K using the ToF spectrometer NEAT at HMI and at T ) 310 and 320 K using the backscattering instrument IRIS at ISIS. The resolutions at the elastic peak position were of 98, 30, and 17.5 µeV (fwhm), respectively. Values for bulk water are given for comparison.30 Figure 3. ToF-derived GDOS for Na0.28CoO21.3H2O (left scale, 4) and dry Na0.7CoO2 (right scale, b) at 295 K. The band observed in Na0.28CoO21.3H2O around 20 meV (indicated with the arrow) can be assigned to water-cation and hydrogen-bond stretch modes.
bond between the water molecule and the lattice23). The data were analyzed using the expressions given in the Appendix. A. GDOS Using NEAT. Figure 3 shows the density of states for Na0.3CoO2‚1.3H2O and Na0.7CoO2 at 295 K obtained using eq 4 in the Appendix. It is clear that, in the spectra of the hydrated sample, the strong peak observed in bulk water around 6 meV is absent. This observation is a distinct fingerprint of water molecules constrained by a matrix,20-22 and demonstrates the effectiveness of the freeze-drying process. Thus we can infer that the measured INS data arise from water confined within the sodium cobaltate crystal structure. Since the local environment of the water molecules consists of two nearest-neighbor sodium atoms, one expects to observe two water-cation stretch modes. Indeed, the observation of a band around 20 meV (indicated by the arrow) is consistent with observations in water mixtures, water in zeolites, and crystal hydrates.23-26 Therefore, as in Na-MFI zeolite,25 which also contains 6 Na+ cations per unit cell, and as in natrolite,23 where the shortest hydrogen-bond length in the crystal is 2.83 Å, the band observed in Na0.28CoO21.3H2 around 20 meV can be assigned to water-cation and hydrogen-bond stretch modes. The out-of-plane vibration of the Co against O, observed at 65 meV as a strong mode in the infrared spectra27 (indicated by *) in Na0.7CoO2, should also be present in the hydrated sample. Thus, its coupling with the librational peak observed in bulk water at about 70 meV28 may mask the shift of the water mode to higher energies. In summary, these results indicate that the water behavior in Na0.3CoO2‚1.3H2O is different from that of bulk water, but consistent with that of confined water. B. Rotational Motions Determined from NEAT Measurements. Figure 4a,b shows typical spectra obtained using the NEAT spectrometer at selected Q values, with energy resolutions of 98 and 30 µeV (fwhm), respectively, at 295 K. Equation 5 of the Appendix was used to analyze these spectra. In the ∆E ) 30 µeV data, the contribution from the fast water molecules is considered within the flat background, while the QENS component is attributed to the dynamics of the slower water molecules. In the energy range analyzed in the present experiment (from -1.5 to 1.5 meV for ∆E ) 98 µeV and from -0.3 to 0.5 meV for ∆E ) 30 µeV), the fitted half width at half-
maximum (hwhm) (Γr) shows a Q-independent quasi-elastic (QE) width broadening, indicating that, in the probed time scale, the protons are undergoing reorientational motions. Γr gives characteristic rotational times of 1.85 and 20 ps (see Figure 4c,d). We now consider the analysis of the experimental elastic incoherent structure factor (EISF), which depends on the spatial distribution of the proton, averaged over a time determined by the instrumental resolution. The variation of the experimental EISF as a function of the momentum transfer, Q, can be compared to different models in order to determine the nature of the rotation. Here we have considered the simplest dynamical models describing rotational jumps (summarized in Table 1), which take the water molecules into orientations allowed by the crystal structure: (i) Isotropic Rotational Diffusion Model: The water molecules are assumed to have, on a time average, no preferred orientation and to perform more or less continuous small-angle random rotations. (ii) Two-Site Jump Model: Ascribes the probability for each H atom to jump between two separated sites parametrized by a jump distance d, between a pair of neighboring O atoms, to which it remains restricted. The experimental EISF for the faster motion is shown as a function of Q in Figure 4e. The lines in the figure show the fit of the experimental EISF as a function of Q using the models described in Table 1. From this analysis, it appears that the twosite jump model has approximately the same variation as the experimental data, so the isotropic rotational diffusion model can be excluded. Also, based on structural arguments,3,9 it is expected that, at 295 K, the water molecules will be chemically bound to the sodium atoms, and one H atom is also bound to the oxygen in the CoO2 sheet, implying rotations around a symmetry axis. The fit parameter p, which gives the fraction of elastic contribution coming from the water molecules bound to the lattice, was found to be 97%, with d ) 2.1 Å. The jump distance, d, corresponds well to the distance between two neighboring H sites in H2O.9 The chemical stability of the SC phase is confirmed by the very small fraction of the water molecules that are mobile in the measured time-scale at 295 K. Furthermore, the faster rotational time is larger than the value for bulk water (1.05 ps29), but similar to the time-characterizing rotation of the water molecules around their C2 axis in intercalated clay minerals.30
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Figure 4. Examples of experimental spectra plotted on a log-scale (O) at selected Q values for Na0.28CoO2‚1.3H2O, (a) ∆E ) 98 µeV (solid line) and (b) ∆E ) 30 µeV (fwhm), together with the best-fit (solid line), the QE components (long dashed lines), and the background (short dashed lines). (c,d) Line-width (hwhm) of the rotational Lorentzian vs Q of the QE component observed in Na0.28CoO2‚1.3H2O at 295 K, τr ) (2p/Γr). (e) The experimental EISF (p) for ∆E ) 98 µeV. The solid line represents the fitted theoretical EISF given in Table 1.
On the other hand, the slower rotation can be attributed to collective planar motions, similar to observations in hydrated Na-vermiculiteclays,nanoporoussilicates,andionicsolutions.31-34 The fitted parameters, characterizing the rotational motions at 295 K, are given in Table 2. C. Freezing of Motion Determined from IRIS Measurements. To probe slower motion in the SC sample, we performed measurements using the IRIS backscattering spectrometer. Initially, scans were carried out of the integrated intensity over the accessible Q range (0.4 e Q e 1.8 Å-1) within an energy window from 0.1 to 0.2 meV as a function of temperature, 5K e T e 275 K (Figure 5). The purpose of such a scan is to detect the onset (or thermal activation) of slow dynamics as the sample is warmed. While, for T e 150 K, the scattering was essentially constant with temperature, in the region between 150 K e T e 230 K, there was an increase of the scattering with increasing temperature. A considerable change in the slope is observed for T g 230 K, similar to the behavior observed in experiments performed in the nanosecond time scale in Na0.3CoO2‚1.4H2O,35 pointing to the presence of water relaxation in addition to the lattice vibrations.36,37 Furthermore, comparing our results with observations in interfacial38 and adsorbed39 water systems, our data can be interpreted as a signature of a proton-glass state below 150 K.
Figure 5. IFW scan of Na0.28CoO2‚1.3H2O measured on IRIS upon heating over the accessible Q range (0.4 e Q e 1.8 Å-1) over the energy window from 0.1 to 0.2 meV. The turn seen above 230 K is due to activation of the water motions. Data are normalized to the monitor. The lines are intended as guide to the eye.
D. Translation Motion Determined from IRIS Measurements above 310 K. Figure 6 shows the results of fits to the high-resolution IRIS data at selected Q values for T ) 320 K using the phenomenological approach given by eq 5 in the
Dynamics of H2O in NaxCoO2‚yH2O
J. Phys. Chem. B, Vol. 112, No. 3, 2008 707
Figure 7. Line-width (hwhm) of the QE component observed in Na0.28CoO2‚1.3H2O as a function of Q2 at 310 and 320 K. The fit to the random jump diffusion observed at 320 K is shown as a solid line.
with the diffusion coefficient Dt given by
Dt )
L2 6τ0
(2)
τ0 is the average residence time between jumps, and L is the mean jump distance. The fit of this model to the data is also shown in Figure 7. Applying eq 5 to the fitted Γ versus Q2 leads to the diffusion coefficient Dt ) 0.9 × 10-9 m2/s and the residence time τ0 ) 22 ps at 320 K in Na0.28CoO2‚1.3H2O. These values are in broad agreement with observations of translational diffusion motion at room temperature for water molecules in the hydration shell of sodium ions.31,41,42 It is important to note that the model parameters are determined as an average over the entire ensemble of water molecules, while the local dynamics of an individual water molecule may depend on the local chemical environment. Figure 6. Examples of experimental spectra (4) at selected Q values for Na0.28CoO2‚1.3H2O, obtained using IRIS with an elastic resolution of 17.5 µeV at 320 K. The spectrum was fitted using eq 3. The solid line is the fitted spectrum, and the dashed line is the QE component. The background is also shown (short dashed line).
Appendix. During the fitting procedure, only one Lorentzian was used, and the background term could be modelled by a straight line. The fitting result is in excellent agreement with the experimental data. To get information about the character of the H-motion, the hwhm of the Lorentzian function Γ was analyzed as a function of Q2. As shown in Figure 7, no clear dependence of the hwhm in Q2 can be detected at 310 K, indicating that only the reorientational motion of the water molecule can be identified. Indeed, the slower rotational time τr(295K) ) 20 ps, mentioned in the previous section, could also be detected at 310 K. Most interesting is a long-range translational diffusion observed at T ) 320 K in the IRIS data indicated by the Q2 dependence of Γ. Subsequently, the translation STinc(Q,ω) can be evaluated in the framework of the random jump diffusion model, where the variation of Γ versus Q2 is approximated by the Singwi and Sjo¨lander model as follows:40
Γ(Q) )
DtQ2 1 + DtQ2τ0
(1)
IV. Discussion It has been shown that the GDOS does not show the behavior expected for bulk water. Similar behavior is observed for the proton weighted density of vibrational states of water in concrete20 and water in Vycor.43 In addition, the frequencies of the water-cation and hydrogen-bond stretch modes are similar to those found in the zeolite natrolite.23 The detected slower rotational motion, τr ) 22 ps, agrees well with reported results in similar systems where Na is surrounded by one hydration layer,31-34 where rotational times of =10-11 s were reported. Accordingly, the water dynamics observed through our QENS experiments in Na0.28CoO2‚1.3H2O fully support the structural data that the water is bound to the lattice. It is noteworthy that the residence time, τ0 = 21 ps, obtained from our QENS data, corresponds well to the correlation time obtained by molecular dynamics (MD) simulations of ions in solution.44 In MD studies, the correlation time is defined as the time over which water molecules of the first coordination shell exchange with molecules in the bulk, and as a consequence any water-ion correlation is lost. On the basis of structural arguments, one expects that, for T e 310 K, the water molecules will be chemically bound to the sodium atoms and one hydroxyl in the CoO2 sheet, implying that the breakup of the first coordination shell is indeed initiated by a molecular reorientation other than translation.45
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The inelastic fixed window (IFW) scan (Figure 5) indicates the onset of a supercooled liquid state38,39 in Na0.28CoO2‚1.3H2O for 150 K e T e 230 K. As similar observations were made in two-dimensional layer-structured Na-vermiculite clay, we hypothesize that the presence of Na+ in sodium cobalt oxyhydrate would also introduce the disorder necessary to prevent total crystallization of the water molecules around the sodium ion in Na0.28CoO2‚1.3H2O.37 Indeed, structural analysis indicates the presence of two conformations for the water molecules around the Na+, while significant diffuse scattering indicates that a glass-like state is also present.9 V. Summary The main findings of our work can be summarized as follows: (i) The GDOS spectrum of Na0.28CoO2‚1.3H2O obtained from NEAT at 295 K is consistent with observations made in systems that present bound water. (ii) The observed proton dynamics are thermally activated above 150 K and are characterized by a fast and slow rotational motion of 1.85 and 20 ps, respectively, for T < 320 K. (iii) As summarized in Table 2, up to 310 K, only rotational motion is observed, but around 320 K, restricted diffusive motion characterized by a diffusion coefficient Dt ) 0.9 × 10-9 m2/s and a residence time τ0 ) 22 ps takes place. Acknowledgment. We have benefited from discussions of this work with A. Desmedt (LPCM, Bordeaux), R. E. Lechner (HMI, Berlin), M. M. Koza (ILL, France), and L. P. Aldridge (ANSTO and UNSW, Australia). We thank D. Alber (HMI, Germany) for the NAA measurements and C. J. Milne for the sample preparation. H.N.B. is thankful to W. Kalceff (UTS, Australia) for helpful comments during the preparation of this manuscript. We also acknowledge the support of the Berlin Neutron Scattering Center (BENSC) and of the ISIS Pulsed Neutron Scattering Facility, Rutherford Appleton Laboratory, in providing the neutron research facilities used in this work. BENSC and ISIS are partners in the EU-supported network of European neutron facilities, the Neutron and Muon Integrated Infrastructure Initiative (NMI3). Appendix The form of the scattering function in the studied energy range is19
Sinc(Q,ω) ) e(-Q 〈uV 〉)[STinc(Q,ω) X SRinc(Q,ω) + 2
2
SIinc(Q,ω)] X R(Q,ω) (3) where SIinc(Q,ω) is the inelastic term, which determines the short-time dynamics of the hydrogen atoms, and contributes only a little in the QE region. STinc(Q,ω) and SRinc(Q,ω) describe the translational and rotational dynamics of the protons, respectively. R(Q,ω) is the instrumental resolution function. The effects of internal molecular and external lattice modes are represented by e(-Q2〈uV2〉). Knowing that the librational modes are particularly sensitive to the local environment of the water, we analyzed the GDOS as16
g(ω) )
2M p2
〈( )(
)
e2W(Q) ω S(Q,ω) Q2 n(ω) + 1
〉
(4)
where M is the mean sample mass, n(ω) is the Bose-Einstein distribution function, e2W is the Debye-Waller factor, and 〈...〉 represents the average over all observed Q values. For the QENS, on which we focus, STinc(Q,ω) and SRinc(Q,ω) are divided into the sum of an elastic component arising from the bound hydrogen plus a QENS component arising from the mobile hydrogen in the time scale accessible to the instrument. Phenomenologically, the self-dynamic structure factor can be described as
S(Q,ω) ) e(-Q 〈uV 〉) [A0(Q)δ(ω) + (1 - A0(Q))Lj(Γj,ω)] X R(Q,ω) + B(Q,ω) (5) 2
2
where the Debye-Waller factor, described by the first term, takes into account the vibrational motions. A0(Q) represents the purely elastic part of the scattering function and gives the number density of the bound (or immobile) hydrogen atoms. The amplitude of the elastic component, A0(Q), commonly called the EISF, provides a measure of the time-averaged spatial distribution of the protons, completely characterizing the geometry of the motion.19 The QE contribution is described by Lorentzian functions Lj(Γj,ω). Their number j, and respective quasi-elastic incoherent structure factor (QISF), (1 - A0(Q)), depend on the particular model used. The parametrized hwhm Γj, or τ-1, involves the different jump-rate probabilities. The background term is described by the function B(Q,ω), and R(Q,ω) gives the resolution function. References and Notes (1) Braconnier, J.-J.; Delmas, C.; Fouassier, C.; Haggenmuller, P. Mater. Res. Bull. 1980, 15, 1797. (2) Terasaki, I.; Sasago, Y.; Uchinokura, K. Phys. ReV. B 1997, 56, R12685. (3) Takada, K.; Sakurai, H.; Takayama-Muromachi, E.; Izumi, F.; Dilanian, R.; Sasaki, T. Nature 2003, 422, 53. (4) Milne, C.; Argyriou, D.; Chemseddine, A.; Aliouane, N.; Veira, J.; Landsgesell, S.; Alber, D. Phys. ReV. Lett. 2004, 93, 247007-1. (5) Roger, M.; Morris, D. J. P.; Tennant, D. A.; Gutmann, M. J.; Goff, J. P.; Hoffmann, J. U.; Feyerherm, R.; Dudzik, E.; Prabhakaran, D.; Boothroyd, A. T.; Shannon, N.; Lake, B.; Deen, P. P. Nature 2007, 445 (7128), 631. (6) Foo, M. L.; Wang, Y.; Watauchi, S.; Zandbergen, H. W.; He, T.; Cava, R. J.; Ong, N. P. Phys. ReV. Lett 2004, 92 (24), 247001. (7) Lynn, J. W.; Huang, Q.; Brown, C. M.; Miller, V. L.; Foo, M. L.; Schaak, R. E.; Jones, C. Y.; Mackey, E. A.; Cava, R. J. Phys. ReV B 2003, 68, 214516. (8) Jorgensen, J. D.; Avdeev, M.; Hinks, D. G.; Short, J. C. B. S. Phys. ReV. B 2003, 68, 214517. (9) Argyriou, D.; Radaelli, P.; Milne, C. J.; Aliouane, N.; Chapon, L.; Chemseddine, A.; Veira, J.; Cox, S.; N. D. Mathur.; Midgley, P. J. Phys.: Condens. Matter 2005, 17, 3293. (10) Jorgensen et al. in ref 8 describe a 1:4 Na-to-H2O coordination on the basis of a disorder model. (11) Moyoshi, T.; Yasui, Y.; Soda, M.; Kobayashi, Y.; Sato, M.; Kakurai, K. J. Phys. Soc. Jpn. 2006, 75, 074705. (12) See, for example, Robinson, R. A.; Stokes, R. Electrolyte Solutions; Butterworths: London, 1955. (13) Jalarvo, N.; Aliouane, N.; Bordallo, H. N.; Milne, C. J.; Veira, J. R.; Argyriou, D. N. Eur. Phys. J. Spec. Top. 2007, 141, 69. (14) Lechner, R. E. Physica B 1992, 180-181, 973. (15) Fitter, J. User Manual for FITMO (NEAT TOF-Data Analysing Programm); HMI: Berlin, 1997. (16) Nipko, J.; Loong, C.-K.; Loewenhaupt, M.; Braden, M.; Reichardt, W.; Boatner, L. Phys. ReV. B 1997, 56, 11584. (17) Carlile, C. J.; Adams, M. A. Physica B 1992, 182, 431. (18) Howells, W. S. Report No. RAL-TR-96-006; Technical Report for Rutherford Appleton Laboratory: Oxfordshire, U.K., 1996. (19) See, for example, M. Be´e. Quasi-Elastic Neutron Scattering; Adam Hilger: Bristol, PA, 1988. (20) Bordallo, H. N.; Aldridge, L. P.; Desmedt, A. J. Phys. Chem. B 2006, 110, 17966. (21) Bellissent-Funel, M.-C.; Teixeira, J. J. Mol. Struct. 1991, 250, 213. (22) Chen, S.-H.; Gallo, P.; Bellissent-Funel, M.-C. Can. J. Phys. 1995, 73, 703.
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